$\dfrac{ -10b - 10c }{ 6 } = \dfrac{ 3b - 5d }{ 5 }$ Solve for $b$.
Solution: Multiply both sides by the left denominator. $\dfrac{ -10b - 10c }{ {6} } = \dfrac{ 3b - 5d }{ 5 }$ ${6} \cdot \dfrac{ -10b - 10c }{ {6} } = {6} \cdot \dfrac{ 3b - 5d }{ 5 }$ $-10b - 10c = {6} \cdot \dfrac { 3b - 5d }{ 5 }$ Multiply both sides by the right denominator. $-10b - 10c = 6 \cdot \dfrac{ 3b - 5d }{ {5} }$ ${5} \cdot \left( -10b - 10c \right) = {5} \cdot 6 \cdot \dfrac{ 3b - 5d }{ {5} }$ ${5} \cdot \left( -10b - 10c \right) = 6 \cdot \left( 3b - 5d \right)$ Distribute both sides ${5} \cdot \left( -10b - 10c \right) = {6} \cdot \left( 3b - 5d \right)$ $-{50}b - {50}c = {18}b - {30}d$ Combine $b$ terms on the left. $-{50b} - 50c = {18b} - 30d$ $-{68b} - 50c = -30d$ Move the $c$ term to the right. $-68b - {50c} = -30d$ $-68b = -30d + {50c}$ Isolate $b$ by dividing both sides by its coefficient. $-{68}b = -30d + 50c$ $b = \dfrac{ -30d + 50c }{ -{68} }$ All of these terms are divisible by $2$ Divide by the common factor and swap signs so the denominator isn't negative. $b = \dfrac{ {15}d - {25}c }{ {34} }$